On the facial structure of the set covering polytope

Abstract

Given a bipartite graphG = (V, U, E), a cover ofG is a subset \(D \subseteq V\) with the property that each nodeu ∈ U is adjacent to at least one nodev ∈D. If a positive weightcv is associated with each nodev ∈V, the covering problem (CP) is to find a cover ofG having minimum total weight.

In this paper we study the properties of the polytopeQ(G) ⊂ ℝ|V|, the convex hull of the incidence vectors of all the covers inG. After discussing some general properties ofQ(G) we introduce a large class of bipartite graphs with special structure and describe several types of rank facets of the associated polytopes.

Furthermore we present two lifting procedures to derive valid inequalities and facets of the polytopeQ(G) from the facets of any polytopeQ(G′) associated with a subgraphG′ ofG. An example of the application of the theory to a class of hard instances of the covering problem is also presented.